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Multiscale modeling of vertebrate limb development

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Abstract We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction–diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin‐to‐limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods
Depictions of the development of normal, experimentally manipulated, and mutant limbs, and of simulations with the TFF model. (Upper box, Left) Proximodistal developmental progression of chicken forelimb between days 3 and 7 of development (indicated by the corresponding Hamburger–Hamilton stages [Hamburger & Hamilton, ]). Early cartilage, including precartilage condensations, shown in light blue; definitive cartilage shown in darker blue. (Right) A series of snapshots from the simulation of normal limb development by the discontinuous Galerkin FEA method (Zhu, Zhang, Alber, & Newman, ), using the morphostatic reduction (Alber et al., ) of the TFF model of Hentschel, Glimm, Glazier, and Newman (). (Lower left box) Simulations of AER removal based on the TFF–FEA model. (Left columns) Drawings of AER removal experiments (Saunders, ). Top images show an intact chicken wing bud at an early stage of development and the limb skeleton that it generates. Middle images show an early stage wing bud with the AER removed, and the resulting limb skeleton, which develops normally but is truncated at the elbow. Bottom images show a later stage wing bud whose AER has been removed. The resulting skeleton is truncated from the wrist onward. (Right column) Simulations of limb development as in upper box. (Top) AER (i.e., the source of suppressive FGF morphogen) active throughout simulation; normal development results. (Middle) AER turned off early during the simulation. (Bottom) AER deleted later during the simulation. All simulations ran for the same duration. (Lower right box) TFF–FEA simulations of effect of distal expansion of developing limb. (Top row, left column) Drawing of expanded chicken wing bud resulting from anterior graft of an ectopic ZPA from another wing bud (normal limb profile at this stage shown in red); (Top row, center column) resulting cartilage skeleton, with mirror‐image duplication; (Top row, right column) End‐stage of simulation with distally expanded limb bud corresponding to that shown on left. (Second row, left column) Expanded mouse forelimb bud in embryos null for both Shh and Gli3 (normal limb profile at this stage shown in red); (Second row, center column) Resulting skeleton, with supernumerary digits (Litingtung, Dahn, Li, Fallon, & Chiang, ); (Second row, right column) End‐stage of simulation with distal expansion corresponding to that shown on left. (Third row, left column) Expanded wing bud of chicken embryo homozygous for talpid2 mutation; (Third row, center column) Cartilage skeleton formed from such a limb bud later during development; (Third row, right column) End‐stage of simulation with distal expansion corresponding to that shown on left. (Bottom row, left column) Shape of the pectoral fin‐bud in an embryo of the dogfish Scyliorhinus torazame; (Bottom row, center column) Cartilaginous fin skeleton formed from such a limb bud; (Bottom row, right column) End‐stage of simulation using a limb bud contour like that shown on left. Source: Figures adapted from Zhu et al. (), which should be referred to for additional details. AER, apical ectodermal ridge; FEA, finite element analysis; FGF, Fibroblast Growth Factor; TFF, TGF‐β, fibronectin, and FGF model; ZPA, zone of polarizing activity
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Tree showing phylogenetic relationship of extant fish groups and hypothesized appearance of patterning mechanisms. (a) Capacity for focal cartilage differentiation. A Gal‐1 capable of mediating mesenchymal aggregation was present in ancestral gnathostomes. With the involvement of products of the conserved toolkit genes BMP and BMP receptor 1, and eventually FGF and FGF receptor 2, in expansion‐restricting roles, these aggregates would have formed focal compactions or protocondensations. If Sox9 was coordinately induced at these sites they would have been converted to cartilage. Fibronectin, under the regulation of the positive autoregulatory TGF‐β, would have consolidated their formation as discrete elements. At some point on this branch of the evolutionary trajectory, a Wnt gene was recruited to the Sox9‐BMP couple, forming the BSW Turing‐type network, and this would have reinforced the counter‐chondrogenic distribution of BMP. Although Gal‐8 was present in these organisms, their CRDs had not yet evolved to compete with those of Gal‐1, so the full 2GL system was not in place. (b) Capacity for formation of nodules, stripes, and plates of cartilage. Evolution of cross‐activation of Gal‐1 and Gal‐8 and of the Gal‐8 CRDs to competitive status with Gal‐1 CRDs created the 2GL system. This generated numerous parallel cartilage rods and, with fusion, plates. (c) Variability of the endoskeleton under positive selection. In the actinopterygians (ray‐finned fishes) all the focal chondrogenesis‐enabling circuitry was carried forward and (depending on the species), possibly the BSW patterning system (although it appears to be absent in the teleosts; Onimaru et al. ()). Selection on the Gal‐8 CRD is hypothesized to have reinforced the 2GL patterning circuitry in some, but not all, actinopterygians. (d) Capacity for regulated proximodistal patterning with small numbers of elements. In the sarcopterygians (lobe‐finned fishes, including tetrapods) the ancestral focal chondrogenesis toolkit and skeletal patterning systems were carried forward, but sometime after their appearance Gal‐8 also acquired a unique conserved noncoding, potentially regulatory motif that is hypothesized to have enabled it to be regulated in a quantitative fashion during elongation of the limb bud. This is hypothesized to be the basis of the evolutionarily robust developmental morphology of the tetrapod limb (Bhat et al., ). Examples of limb skeletons typical of the various groups are indicated on the respective branches. Source: Based, with modifications, on Newman et al. (), fig. 5). CRDs, carbohydrate‐recognition domains; 2GL, Two Galectin + Ligands
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Dependence of the 2GL skeletal pattern formation model on various parameters. (Left box) Plots of the Gal‐1‐dependent spatial cell density for the full 2GL model at times t = 0 (dashed line) and t = 1 (solid line), for different values of the cell–cell adhesion constant αK. For sufficiently small αK, no spatial patterns are produced. As αK is increased, periodic patterns appear, with an abrupt transition between 200 < αK < 250. Further increase in αK leads to fine‐tuning of the pattern. (See Glimm et al. () for details.) (Right box) Gal‐8 dependence of pattern formation in the condensation‐permissive parameter space of the reduced 2GL model. Two‐parameter bifurcation diagram representing the dependence of condensation patterns on μ, the expression rate of Gal‐8, shown on the vertical axis, and Gal‐8 binding affinity β shown on the horizontal axis. A single plane of the complete parameter space is shown in which the concentration values of Gal‐1 are compatible with condensation formation. According to the model, Gal‐8 can participate in skeletogenic interactions with Gal‐1 only if it is capable of reversibly competing with the condensation‐promoting role of Gal‐1. This capacity, reflected in the values of β, is a function of the primary structure of Gal‐8, several residues of which in sarcopterygians have been subject to purifying selection. The expression rate of Gal‐8, reflected in μ, is either constitutive or developmentally regulated, and Gal‐8 genes of sarcopterygians, including the tetrapods, have acquired potential regulatory sequences for proximodistally expressed transcription factors. These could have enabled the proximodistal changes in the number of parallel elements with limb elongation seen in these species. Approximate contours demarcating the calculated number of distinct condensations are shown via a heat map within the condensation region. Source: Graph taken from Bhat, Chakraborty, Glimm, Stewart, and Newman (), with computations based on the model described in Glimm et al. (). 2GL, Two Galectin + Ligands
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Venn diagram of the categories of models discussed in this review. The main models involve (a) local self‐organization of mesenchymal cells into precartilage condensations (e.g., Zeng et al., ; Zeng, Thomas, & Glazier, ), or variations of this tissue mechanics scheme in which tensile forces due to long‐range organization of the ECM influence the global character of the patterns (e.g., Oster et al., ); (b) interactions of morphogen production and transport leading to chemical patterns for eventual placement of condensations (e.g., Badugu, Kraemer, Germann, Menshykau, & Iber, ; Hentschel et al., ; Newman & Frisch, ; Raspopovic et al., ), or variations of this reaction‐diffusion scheme in which the specific characters of patterned skeletal elementsare influenced by monotonic PI gradients (e.g., Glimm, Zhang, Shen, & Newman, 2012; Raspopovic et al., 2014); and (c) mechanisms in which tissue mechanics and RD effects interact in a morphodynamic fashion (sensu Salazar‐Ciudad et al., ) in the generation of skeletal elements (e.g., Glimm et al., ; Oster, Murray, & Maini, ) or variations on this hybrid scheme in which coordination and refinement of skeletal elements is effected by global synchronization of cellular oscillators (Glimm et al., 2019). ECM, extracellular matrix; PI, positional information; RD, reaction–diffusion; TM, tissue mechanics
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Analytical and Computational Methods > Analytical Methods
Developmental Biology > Developmental Processes in Health and Disease
Models of Systems Properties and Processes > Mechanistic Models
Analytical and Computational Methods > Computational Methods

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